Quantum Fisher information and entanglement

Transcription

1 1 / 70 Quantum Fisher information and entanglement G. Tóth 1,2,3 1 Theoretical Physics, University of the Basque Country (UPV/EHU), Bilbao, Spain 2 IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 3 Wigner Research Centre for Physics, Budapest, Hungary Quantum Information Theory and Mathematical Physics, BME, Budapest, September, 2018

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3 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 3 / 70

4 Motivation The quantum Fisher information (QFI) plays a central role in metrology. In linear interferometers, the QFI is directly related to multipartite entanglement. Thus, one can detect entanglement with precision measurements.

5 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 5 / 70

6 Entanglement A state is (fully) separable if it can be written as p k ϱ (1) k ϱ (2) k... ϱ (N) k. k If a state is not separable then it is entangled (Werner, 1989).

7 k-producibility/k-entanglement A pure state is k-producible if it can be written as Φ = Φ 1 Φ 2 Φ 3 Φ 4... where Φ l are states of at most k qubits. A mixed state is k-producible, if it is a mixture of k-producible pure states. e.g., Gühne, GT, NJP If a state is not k-producible, then it is at least (k + 1)-particle entangled. two-producible three-producible

8 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 8 / 70

9 Quantum metrology Fundamental task in metrology ϱ U (θ )=exp( iaθ ) ϱ θ We have to estimate θ in the dynamics U = exp( iaθ).

10 The quantum Fisher information Cramér-Rao bound on the precision of parameter estimation ( θ) 2 1 F Q [ϱ, A], ( θ) 2 F Q [ϱ, A]. where F Q [ϱ, A] is the quantum Fisher information. The quantum Fisher information is F Q [ϱ, A] = 2 k,l (λ k λ l ) 2 k A l 2 := F λ k + λ Q (ϱ; i[ϱ, A]), l where ϱ = k λ k k k.

11 Special case A = J l The operator A is defined as A = J l = N n=1 j (n) l, l {x, y, z}. Magnetometry with a linear interferometer B B Δϴ sn Δϴ ss y y z x z x

12 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 12 / 70

13 The quantum Fisher information vs. entanglement For pure product states of N spin- 1 2 particles we have Ψ = ψ (1) ψ (2) ψ (3)... ψ (N), F Q [ϱ, J z ] = 4( J z ) 2 = 4 N n=1 ( j (n) z ) 2 N. For separable states (mixtures of pure product states) we have since F Q [ϱ, A] is convex in ϱ. F Q [ϱ, J l ] N, l = x, y, z, Pezze, Smerzi, Phys. Rev. Lett. 102, (2009); Hyllus, Gühne, Smerzi, Phys. Rev. A 82, (2010).

14 The quantum Fisher information vs. multipartite entanglement For N-qubit k-producible states states, the quantum Fisher information is bounded from above by where n is the integer part of N k. If k is divisor of N then F Q [ϱ, J l ] nk 2 + (N nk) 2. F Q [ϱ, J l ] kn. P. Hyllus et al., Phys. Rev. A 85, (2012); GT, Phys. Rev. A 85, (2012).

15 The quantum Fisher information vs. macroscopic superpositions Macroscopic superpositions (e.g, GHZ states, Dicke states) F Q [ϱ, J l ] N 2, F. Fröwis and W. Dür, New J. Phys (2012).

16 Summary of various types of limits Bounds for the QFI Shot-noise limit: F Q [ϱ, J l ] N, Heisenberg limit: F Q [ϱ, J l ] N 2. Bounds for the precision Shot-noise limit: ( θ) 2 1 N, Heisenberg limit: ( θ) 2 1 N 2.

17 Scaling of the precision in a noisy environment Is the scaling F Q [ϱ, J l ] N 2 possible? Too good to be true? One feels that this is probably not possible. Due to uncorrelated local noise the scaling returns to the shot-noise scaling F Q [ϱ, J l ] const. N R. Demkowicz-Dobrzański J. Kołodyński, M. Guţǎ, Nat. Commun. 3, 1063 (2012); B. Escher, R. de Matos Filho, L. Davidovich, Nat. Phys. 7, 406 (2011).

18 Entanglement detection with precision measurement Entanglement detection in cold gases. B. Lücke et al., Science, Science 334, 773 (2011).

19 Entanglement detection with precision measurement II Entanglement in a photonic experiment. Krischek et al., Phys. Rev. Lett. 107, (2011)

20 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 20 / 70

21 Families of the Quantum Fisher information Definition The quantum Fisher information is defined as where ˆF f (ϱ; A) = Tr(AJ 1 f (ϱ)a), and J f ϱ(a) = f (L ϱ R ϱ 1 )R ϱ, L ϱ (A) = ϱa, R ϱ (A) = Aϱ. f : R + R + is a standard operator monotone function, which has the properties xf (x 1 ) = f (x) and f (1) = 1. Mean based on f m f (a, b) = af ( ) b. a

22 Families of the Quantum Fisher information II Form with density matrix eigenvalues and eigenvectors ˆF f Q (ϱ; A) = i,j 1 m f (λ i, λ j ) i A j 2.

23 The ususal QFI, ˆF Q (ϱ; A) F Q [ϱ, A] For the arithmetic mean m f (a, b) = a+b 2, ˆF Q (ϱ; A) = i,j 2 λ i + λ j A ij 2. ˆF(ϱ; A) is the smallest among the various types of the generalzied quantum Fisher information. [Normalization f(1)=1.] The quantum Fisher information is defined for the linear dynamics ϱ output (t) = ϱ + At Cramer Rao bound: ( t) 2 1/ˆF (ϱ; A). D. Petz, J. Phys. A: Math. Gen. 35, 929 (2002); P. Gibilisco, F. Hiai, and D. Petz, IEEE Trans. Inform. Theory 55, 439 (2009).

24 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 24 / 70

25 Families of the variances Definition Generalized variance var ˆ f ϱ(a) = A, J f ϱ(a) (TrϱA) 2, where f is the matrix monotone function mentioned before. For the arithmetic mean m f (a, b) = a+b 2, we get the usual variance var ˆ f ϱ(a) = A 2 ϱ A 2 ϱ. It is the largest among the generalized variances.

26 Families of the variances II Form given with the density matrix eigenvalues and eigenvectors var f ϱ(a) = m f (λ i, λ j ) A ij 2 2 λ i A ii. i,j The usual variance is the largest. (The arithmetic mean is the largest mean.) For pure states, they give 2m f (1, 0) ( A) 2.

27 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 27 / 70

28 Family of the Quantum Fisher information for unitary dynamics 1 In physics, we are typically interested in a unitary dynamics ϱ output (t) = exp( iaθ)ϱ exp(+iaθ), 2 Then, ˆF f Q [ϱ, A] = ˆF f (ϱ; i[ϱ, A]). 3 We propose the normalization FQ f [ϱ, A] = 2m f (1, 0)ˆF Q f [ϱ, A] 4 For a pure state Ψ we have F f Q [ϱ, A] = 4( A)2 Ψ.

29 Family of the Quantum Fisher information for unitary dynamics II The generalized QFIs given with the density matrix eigenvalues F f Q [ϱ, A] = 2 i,j m f (1, 0) m f (λ i, λ j ) (λ i λ j ) 2 A ij 2. For pure states, equals 4( A) 2. The usual QFI is the largest. (The arithmetic mean is the largest mean.) Exactly the opposite of what we had with the original normalization!

30 The ususal QFI, unitary dynamics For the arithmetic mean m f (a, b) = a+b 2, F Q [ϱ, A] = i,j 2 λ i + λ j (λ i λ j ) 2 A ij 2. "THE" quantum Fisher Information. The quantum Fisher information is defined for the linear dynamics ϱ output (t) = e iaθ ϱe +iaθ. Cramer Rao bound: ( θ) 2 1/F [ϱ, A].

31 Family of the Quantum Fisher information for unitary dynamics IV Definition Generalized quantum Fisher information F Q [ϱ, A] 1 For pure states, we have F Q [ Ψ Ψ, A] = 4( A Ψ ) 2. The factor 4 appears to keep the consistency with the existing literature. 2 For mixed states, F Q [ϱ, A] is convex in the state.

32 Family of variances 1 We propose the normalization var f ϱ(a) = varf ˆ ϱ(a) 2m f (1, 0). 2 For a pure state Ψ we have var f Ψ (A) = ( A)2 Ψ.

33 Family of variances II Form with density matrix eigenvalues var f ϱ(a) = 1 m f (λ i, λ j ) 2 m f (1, 0) A ij 2 2 λ i A ii. i,j For pure states, equals ( A) 2. The usual variance is the smallest. Exactly the opposite of what we had with the original normalization!

34 Family of variances III Definition The generalized variance var ϱ (A) is defined by the following two requirements. 1 For pure states, the generalized variance equals the usual variance var Ψ (A) = ( A) 2 Ψ. 2 For mixed states, var ϱ (A) is concave in the state.

35 Families II QFI: original, linear dynamics, by D. Petz ˆF f (ϱ; A) usual QFI is smallest unitary dynamics ˆF f [ϱ, A] = ˆF f (ϱ; i[ϱ, A]) - unitary dynamics, our normalization FQ f [ϱ, A] = 2m f (1, 0)ˆF Q f [ϱ, A] usual QFI F Q [ϱ, A] is largest variance: original, by D. Petz our normalization var ˆ f ϱ(a) var f ϱ(a) = ˆ usual variance is largest varf ϱ (A) 2m f (1,0) usual variance is smallest

36 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 36 / 70

37 The QFI as a convex roof We have a family of generealized QFI s, which are convex in the state and all equal to four times the variance for pure states. There is a smallest of such functions defined by the convex roof. The usual QFI equals F Q [ϱ, A] = 4 inf {p k,ψ k } p k ( A) 2 Ψ k, k where ϱ = k p k Ψ k Ψ k. GT, D. Petz, Phys. Rev. A 87, (2013); S. Yu, arxiv (2013).

38 The variance as a concave roof We have a family of generealized variances, which are concave in the state and all equal to the variance for pure states. There is a largest of such functions defined by the concave roof. The usual variance is ( A) 2 ϱ = sup p k ( A) 2 Ψ k, {p k,ψ k } k where ϱ = k p k Ψ k Ψ k. GT, D. Petz, Phys. Rev. A 87, (2013).

39 Summary of statements Decompose ϱ as ϱ = k p k Ψ k Ψ k. Then, 1 4 F Q[ϱ, A] k p k ( A) 2 Ψ k ( A) 2 ϱ holds. Both inequalities can be saturated by some decompostion..

40 Variance For 2 2 covariance matrices there is always {p k.ψ k } such that C ϱ = sup {p k,ψ k } p k C Ψk, [Z. Léka and D. Petz, Prob. and Math. Stat. 33, 191 (2013)] k For 3 3 covariance matrices, this is not always possible. Necessary and sufficient conditions for an arbitrary dimension. [D. Petz and D. Virosztek, Acta Sci. Math. (Szeged) 80, 681 (2014)]

41 Why convex and concave roofs are interesting? Convex and concave roofs appear in entanglement theory (E.g., Entanglement of Formation). They do not often appear in other fields. Expression with convex and concave roofs typically cannot be computed with a single formula. The variance and the QFI are defined via roofs, but they can easily be calculated.

42 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 42 / 70

43 Witnessing the quantum Fisher information based on few measurements The bound based on w = Tr(ϱW ) is given as [ ] F Q [ϱ, J z ] sup rw Fˆ Q (rw ). r The Legendre transform is Optimization over ϱ: complicated. ˆF Q (W ) = sup( W ϱ F Q [ϱ, J z ]). ϱ Due to the properties of F Q mentioned before, it can be simplified [ ˆF Q (W ) = sup {λ max W 4(J z µ) 2]}. µ Optimziation over a single parameter! I. Apellaniz, M. Kleinmann, O. Gühne, and G. Tóth, Phys. Rev. A 95, (2017), Editors Suggestion.

44 Example: bound based on fidelity Let us bound the quantum Fisher information based on some measurements F Q N 2 (1 2F GHZ ) FQ[, J z]/f m ax Q if F GHZ > 1 2 FQ[, J y]/f m ax Q F GHZ F D icke Quantum Fisher information vs. Fidelity with respect to (a) GHZ states and (b) Dicke states for N = 4, 6, 12. Apellaniz et al., Phys. Rev. A 2017

45 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 45 / 70

46 QFI as a convex roof used for numerics The optimization over convex decompostions for F Q [ϱ, A] can be rerwrittem as ( F Q [ϱ, A] = 4 A 2 ϱ sup p k A 2 Ψ k ), {p k, Ψ k } where {p k, Ψ k } refers to a decomposition of ϱ. Can be rewritten as an optimization over symmetric separable states F Q [ϱ, A] = 2 inf ϱ ss S s, Tr 1 (ϱ ss)=ϱ k (A 1 1 A) 2 ϱss, where S s is the set of symmetric separable states. States in S s are mixtures of symmetric product states, i.e., they are of the form p k Ψ k Ψ k 2. k Every symmetric separable state can be written in this form.

47 QFI as a convex roof used for numerics II Set of symmetric states with a positive definite partial transpose S SPPT S s. Lower bound on the quantum Fisher information F Q [ϱ, A] inf ϱ SPPT S SPPT, Tr 1 (ϱ SPPT )=ϱ (A 1 1 A) 2 ϱsppt. The bound can be calculated with semidefinite programming. It is a lower bound, not an upper bound! Similar ideas can be used to look for a lower bound on the QFI for given operator expectation values, or compute other convex-roof quantities (e.g., linear entropy of entanglement). GT, T. Moroder, O. Gühne, PRL 2015; see also M. Christandl, N. Schuch, A. Winter, PRL 2010.

48 Outline 1 Motivation Motivation 2 Basics of Entanglement Entanglement 3 Entanglement condition with the Quantum Fisher information Which quantum Fisher information is it? Entanglement condition based on QFI 4 Families of the Quantum Fisher information and variance Families of the Quantum Fisher information Families of the variances Families for unitary dynamics and another normalization Convex roofs and concave roofs 5 Results related to the QFI being a convex roof Estimating the QFI using the Legendre transform Estimating the QFI with semidefinite programming Bounding the quantum Fisher information based on the variance 48 / 70

49 Bound based on the variance Let us define the quantity V (ϱ, A) := ( A) F Q[ϱ, A]. It is well known that V (ϱ, A) = 0 for pure states. For states sufficiently pure V (ϱ, A) is small. For states that are far from pure, the difference can be larger.

50 Generalized variance Generalized variances (with "old" normalization) are defined as ˆ var f ϱ(a) = ij m f (λ i, λ j ) A ij 2 ( λi A ii ) 2, where f : R + R + is a standard matrix monotone function, and m f (a, b) = bf (b/a) is a corresponding mean. Petz, J. Phys. A 35, 929 (2002); Gibilisco, Hiai, and Petz, IEEE Trans. Inf. Theory 55, 439 (2009). The f (x) are bounded as f min (x) f (x) f max (x), where f min (x) = 2x 1 + x, f max(x) = 1 + x 2.

51 Generalized variance II The generalized variance with f (x) = f max (x) is the usual variance (m f is the arithmetic mean.) var ˆ max ϱ (A) = A 2 A 2. Let us now consider the generalized variance with f min (x) m min (a, b) = 2ab/(a + b). Then, (m f is the harmonic mean.) var ˆ min (A) V (ϱ, A). V is the smallest among the generalized varinces V (ϱ, A) = ( A) F Q[ϱ, A] ϱ var ˆ f ϱ(a) ( A) 2.

52 Bound based on the variance, rank-2 Observation For rank-2 states ϱ, ( A) F Q[ϱ, A] = 1 2 [1 Tr(ϱ2 )]( σ 1 σ 2 ) 2 holds, where σ k are the nonzero eigenvalues of the matrix A kl = k A l. Here k are the two eigenvectors of ϱ with nonzero eigenvalues. Thus, σ k are the eigenvalues of A on the range of ϱ. Note S lin (ϱ) = 1 Tr(ϱ 2 ) = 1 k λ 2 k = k l λ k λ l. G. Tóth, arxiv:

53 Bound based on the variance, arbitrary rank Observation For states ϱ with an arbitrary rank we have ( A) F Q[ϱ, A] 1 2 S lin(ϱ) [σ max (A) σ min (A)] 2, where σ max (X) is the largest eigenvalue of X. Estimate F Q : 1 Measure the variance. 2 Estimate the purity. 3 Find a lower bound on F Q. G. Tóth, arxiv:

54 Bound based on the variance, arbitrary rank II Proof. V can also be defined as a concave roof V (ϱ, A) = ( A) F Q[ϱ, A] = sup p k ( A Ψk A ) 2. {p k,ψ k } We want to show that sup p k ( A Ψk A ) S lin(ϱ) [σ max (A) σ min (A)] 2. {p k,ψ k } k Our relation is true, if and only if ( ) X := 1 2 S lin p k Ψ k Ψ k [σ max (A) σ min (A)] 2 k k p k ( A Ψk A ) 2 k is non-negative for all possible choices for p k and Ψ k.

55 Bound based on the variance, arbitrary rank III Minimize X over p = (p 1, p 2, p 3,...) under the constraints p k 0, k p k = 1, while keeping the Ψ k fixed. Further constraint: A = k p k A Ψk = A 0, where A 0 is a constant. X is a concave function of p k s. Hence, it takes its minimum on the extreme points of the convex set of the allowed values for p. For the extreme points, at most two of the p k s are non-zero. Thus, we need to consider rank-2 states only, for which the statement is true due to previous observation ( A) F Q[ϱ, A] = 1 2 [1 Tr(ϱ2 )]( σ 1 σ 2 ) 2.

56 Bound based on the variance, numerical test ( A) F Q[ϱ, A] 1 2 S lin(ϱ)[σ max (A) σ min (A)] 2. (7)

57 Summary We discussed that the quantum Fisher information can be defined as a convex roof of the variance. We also discussed, hoe the quantum Fisher information is connected to quantum entanglement. THANK YOU FOR YOUR ATTENTION!

58 Appendix The variance as a convex roof

59 The variance as a convex roof The usual variance equals four times this concave roof ( A) 2 ϱ = sup p k ( A) 2 Ψ k, {p k,ψ k } k where ϱ = k p k Ψ k Ψ k. GT, D. Petz, Phys. Rev. A 87, (2013).

60 The variance as a convex roof II Decomposition of the density matrix ϱ = k p k Ψ k Ψ k. For all decompositions { p k, Ψ k } ( A) 2 ϱ max p k ( A) 2 Ψ {p k, Ψ k } k k k Important property of the variance: p k ( A) 2 Ψ k. ( A) 2 ϱ = k p k [( A) 2 Ψ k + ( A Ψ k A ϱ ) 2]. If for ϱ there is a decomposition { p k, Ψ k } such that the subensemble expectation values equal the expectation value for the entire ensemble (i.e., A Ψ k = A ϱ for all k) then ( A) 2 ϱ = max p k ( A) 2 Ψ {p k, Ψ k } k = p k ( A) 2. Ψk k k In this case, for ϱ, the usual variance ( A) 2 ϱ is the concave roof of the variance.

61 The variance as a convex roof III Lemma 1 For any rank-2 ϱ there is such a decompositions { p k, Ψ k }. Eigendecomposition of the state ϱ ϱ = p Ψ 1 Ψ 1 + (1 p) Ψ 2 Ψ 2. We define now the family of states Ψ φ = p Ψ p Ψ 2 e iφ. Expectation value of the operator A Ψ φ A Ψ φ = A ϱ + 2 p(1 p)re Clearly, there is an angle φ 1 such that ( ) Re Ψ 1 A Ψ 2 e iφ 1 = 0. For this angle ( Ψ 1 A Ψ 2 e iφ). Ψ φ A Ψ φ = Ψ φ+π A Ψ φ+π = A ϱ.

62 The variance as a convex roof IV In the basis of the states Ψ 1 and Ψ 2, we can write the projection operators onto Ψ φ1 as Ψ φ1 Ψ φ1 [ = p p(1 p)e +iφ 1 p(1 p)e iφ 1 1 p ]. Ψ φ1 +π Ψ φ1 +π [ = p p(1 p)e iφ 1 p(1 p)e +iφ 1 1 p ]. ϱ can be decomposed as We proved Lemma 1. ϱ = 1 2 ( Ψ φ 1 Ψ φ1 + Ψ φ1 +π Ψ φ1 +π ).

63 The variance as a convex roof V Lemma 2 Eigendecomposition of a density matrix ϱ 0 = r 0 k=1 λ k Ψ k Ψ k with all λ k > 0. Rank of the density matrix as r(ϱ 0 ) = r 0, r 0 3. Define A 0 as A 0 = Tr(Aϱ 0 ). We claim that for any A, ϱ 0 can always be decomposed as ϱ 0 = pϱ + (1 p)ϱ +, such that r(ϱ ) < r 0, r(ϱ + ) < r 0, and Tr(Aϱ + ) = Tr(Aϱ ) = A 0. For the proof, see GT, D. Petz, Phys. Rev. A 87, (2013).

64 The variance as a convex roof VI λ 3 1 ρ ρ - ρ + 1 λ 2 1 λ 1 Figure: The rank-3 mixed state ϱ 0 is decomposed into the mixture of two rank-2 states, ϱ and ϱ +. From Lemma 1 and Lemma 2, the main statement follows ( A) 2 ϱ = inf p k ( A) 2 Ψ {p k,ψ k } k. k

65 Appendix Quantities Averaged over SU(d) generators

66 Quantities Averaged over SU(d) generators Any trcaeless Hermitian operator with Tr(A 2 ) = 2 can be obtained as A n := A T n, where A = [A (1), A (2), A (3),...] T, n is a unitvector with real elements, (.) T is matrix transpose. We define the average over unit vectors as f = f ( n)m(d n), M(d n) We would like to compute average of V for operators. It is zero only for pure states. Similar to entropies.

67 Bound on the average V Observation The average of V over traceless Hermitian matrices with a fixed norm is given as V (ϱ) = 2 [ ] d 2 S lin (ϱ) + H(ϱ) 1, 1 where d is the dimension of the system, and H(ϱ) = 2 k,l λ k λ l λ k + λ l = k l λ k λ l λ k + λ l.

68 Average quantum Fisher information The average of the quantum Fisher information can be obtained as F Q [ϱ] = 8 N g [d H(ρ)]. It is maximal for pure states.

69 Bound based on the variance II H( ) exp[s( )] H( ) exp[s( )] Figure: The relation between the von-neumann entropy and H(ϱ) for d = 3 and 10. (filled area) Physical quantum states. (dot) Pure states. (square) Completely mixed state. We see that H(ϱ) exp[s(ϱ)].

70 Other type of quantum Fisher information The alternative form of the usual quantum Fisher information is defined as d 2 d 2 θ S(ϱ e iaθ ϱe +iaθ ) θ=0 = F log Q [ϱ, A]. With that F log Q [ϱ] = 2 N g ( 2dS + 2 k log λ k ).